A factorial (written as n!) is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials grow extremely fast — 10! = 3,628,800 and 20! ≈ 2.43 × 10¹⁸. They appear in combinatorics, probability, calculus (Taylor series), and statistics. The factorial function is only defined for non-negative integers.

How do you calculate 10 factorial?

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800. You can also compute it step by step: 1!=1, 2!=2, 3!=6, 4!=24, 5!=120, 6!=720, 7!=5040, 8!=40320, 9!=362880, 10!=3628800. A useful property: n! = n × (n-1)!, so each factorial is just the previous one multiplied by n.

What is the difference between permutations and combinations?

Permutations (nPr) count ordered arrangements: the order matters. Combinations (nCr) count unordered selections: the order does not matter. For example, choosing 3 letters from {A, B, C}: the combinations are just {A,B,C} = 1 way, but the permutations are ABC, ACB, BAC, BCA, CAB, CBA = 6 ways. nPr = n! / (n−r)! and nCr = n! / (r!(n−r)!). Since order matters for permutations, nPr ≥ nCr always.

What is 0! and why does it equal 1?

0! = 1 by definition, and this is consistent with several mathematical reasons. First, the recursive formula n! = n × (n-1)! requires 1! = 1 × 0!, so 0! must equal 1. Second, the number of ways to arrange 0 objects is exactly 1 (there is one empty arrangement). Third, this definition makes formulas like nCr work correctly at the boundaries — C(n,0) = n!/(0!·n!) = 1, meaning there is exactly one way to choose nothing from n items.

How large can factorials get?

Factorials grow faster than exponentials. 100! ≈ 9.33 × 10¹⁵⁷ — a number with 158 digits. 170! ≈ 7.26 × 10³⁰⁶. Beyond 170!, standard 64-bit floating-point numbers overflow to Infinity. The number of atoms in the observable universe is estimated at about 10⁸⁰, so 57! already exceeds that. For large n, use Stirling's approximation: n! ≈ √(2πn)(n/e)ⁿ.

What is Stirling's approximation?

Stirling's approximation provides an efficient estimate for large factorials without computing the full product: n! ≈ √(2πn) × (n/e)ⁿ, where e ≈ 2.71828. For n = 10: √(20π) × (10/e)¹⁰ ≈ 7.926 × 3,678,794 ≈ 3,598,695 — close to the exact 3,628,800 (about 0.8% error). The relative error decreases as n increases, making it excellent for large n. It is used in statistical physics and combinatorics analysis.

What are real-world uses of factorials?

Factorials power combinatorics — the math of counting possibilities. In probability: if you shuffle a deck of 52 cards, there are 52! ≈ 8 × 10⁶⁷ possible orderings — more than the number of atoms in the observable universe. In cryptography, factorials determine key space sizes. In chemistry, factorials count molecular arrangements. Taylor series expansions (sin x, cos x, eˣ) use factorials in denominators to ensure convergence. Any time you ask "how many ways can n things be arranged?", factorials are the answer.

How many ways can you arrange 10 objects?

10! = 3,628,800 ways. This is called the number of permutations of 10 distinct objects. If only 3 of the 10 positions are being filled (nPr with n=10, r=3), the count drops to 10!/(10-3)! = 10!/7! = 720. If the 3 chosen objects can be in any order without distinction (nCr), it falls further to 10!/(3!×7!) = 120. Each step applies a factorial division to remove overcounting.

Factorial Calculator — n!, nPr, nCr

Calculates n! (factorial), nPr (permutations), and nCr (combinations) for any non-negative integer, with Stirling's approximation for large values.

How to Use the Factorial Calculator

This factorial calculator computes n! instantly for any non-negative integer n. To also compute permutations (nPr) and combinations (nCr), enter the value of r as well. Results for n > 20 are shown in scientific notation since exact values exceed JavaScript's safe integer range. The calculator also shows Stirling's approximation for n ≥ 10 to illustrate the growth pattern.

For example, enter n = 52 to find the number of possible orderings of a standard deck of cards: 52! ≈ 8.07 × 10⁶⁷. This number is so large that if every human who ever lived had been shuffling 1000 decks per second since the Big Bang, they would not have exhausted all orderings. For related topics, see our proportion calculator.

Factorial Formula and Properties

The factorial function is defined recursively:

0! = 1 (by definition)
n! = n × (n − 1)! for n ≥ 1

Key factorials to know: 1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 10! = 3,628,800, 20! ≈ 2.43 × 10¹⁸, 100! ≈ 9.33 × 10¹⁵⁷.

Permutations — nPr

The number of ordered arrangements of r items chosen from n distinct items:

nPr = n! / (n − r)!

Example: How many ways can you arrange 3 books from a shelf of 8? P(8,3) = 8! / 5! = 8 × 7 × 6 = 336 ordered arrangements.

Combinations — nCr

The number of unordered selections of r items from n distinct items (order does not matter):

nCr = n! / (r! × (n − r)!)

Example: How many ways can you choose a committee of 3 from 8 people? C(8,3) = 8! / (3! × 5!) = 56. This is the binomial coefficient, fundamental to probability and Pascal's triangle.

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Stirling's Approximation

For large n, computing n! by multiplying every integer from 1 to n is impractical. Stirling's approximation provides an excellent estimate:

n! ≈ √(2πn) × (n/e)ⁿ

Where e = 2.71828... (Euler's number). The approximation becomes more accurate as n grows — the relative error is about 1/(12n), meaning less than 1% error for n > 8. Stirling's formula is essential in statistical mechanics (calculating entropy), information theory, and asymptotic analysis of algorithms.

Real-World Applications of Factorials

Card games — a 52-card deck has 52! ≈ 8 × 10⁶⁷ possible orderings; every shuffle is almost certainly unique in history
Probability — the denominator of many probability formulas involves factorials through combinations
Taylor series — the expansion of eˣ = Σ(xⁿ/n!) uses factorials in every term
Combinatorics — counting passwords, PINs, combinations of ingredients in recipes
Statistics — the binomial distribution P(X=k) = C(n,k) × pᵏ × (1−p)^(n−k) uses combinations

For calculations involving large numbers in other formats, our decimal calculator handles precision arithmetic.

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Common Factorial Values Table

0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5,040
8! = 40,320
9! = 362,880
10! = 3,628,800
15! = 1,307,674,368,000
20! ≈ 2.43 × 10¹⁸
50! ≈ 3.04 × 10⁶⁴
100! ≈ 9.33 × 10¹⁵⁷

Sources & References

Factorial — Wikipedia
Permutation — Wikipedia
Combination — Wikipedia
Stirling's approximation — Wikipedia

Factorial Calculator